I am using Lightroom's flash galleries within my site, and have been adding the copyright stamp to images. However, the stamp appears to be applied in the gallery and not on the image, so it is possible for internet users to copy the images without the stamp.

Is there an easy way of adding the copyright stamp to each image without producing a separate set and editing in Photoshop?

Are you sure the copyright imprint isn't being applied to the images? Which flash gallery are you using? With the "Lightroom Flash Gallery", the copyright imprint IS applied to each generated JPEG (if you select the "Add Copyright Watermark" tickbox of course).